Polonium. In the 1910 article “The Probability Variations in the Distribution of α Particles” (Philosophical Magazine, Series 6, No. 20, pp. 698–707), E. Rutherford and H. Geiger described the results of experiments with polonium. The experiments indicate that the number of α (alpha) particles that reach a small screen during an 8-minute interval has a Poisson distribution with parameter λ = 3.87. Determine the probability that, during an 8-minute interval, the number, Y, of α particles that reach the screen is a. exactly four. b. at most one. c. between two and five, inclusive. d. Construct a table of probabilities for the random variable Y . Compute the probabilities until they are zero to three decimal places. e. Draw a histogram of the probabilities in part (d). f. On average, how many alpha particles reach the screen during an 8-minute interval?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Polonium. In the 1910 article “The Probability Variations in the Distribution of α Particles” (Philosophical Magazine, Series 6, No. 20, pp. 698–707), E. Rutherford and H. Geiger described the results of experiments with polonium. The experiments indicate that the number of α (alpha) particles that reach a small screen during an 8-minute interval has a Poisson distribution with parameter λ = 3.87.
Determine the probability that, during an 8-minute interval, the number, Y, of α particles that reach the screen is
a. exactly four.
b. at most one.
c. between two and five, inclusive.
d. Construct a table of probabilities for the random variable Y . Compute the probabilities until they are zero to three decimal places.
e. Draw a histogram of the probabilities in part (d).
f. On average, how many alpha particles reach the screen during an 8-minute interval?
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